The generator matrix

 1  0  1  1  1  1  1  1  1  0  1  1  1
 0  1  1  a a^2  0 a^2*X+1  a a^2*X+a^2  1 a^2*X+1 a^2*X+a^2  0
 0  0 a^2*X  0  0  X  0 a*X a*X a^2*X a^2*X a*X  X
 0  0  0  X  0 a^2*X a*X a^2*X  0 a^2*X a^2*X a^2*X  X
 0  0  0  0 a^2*X  X  X a^2*X a*X a*X  X  0 a^2*X

generates a code of length 13 over F4[X]/(X^2) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+84x^28+144x^30+165x^32+384x^33+1296x^34+180x^36+2304x^37+3888x^38+240x^40+3456x^41+3888x^42+204x^44+138x^48+12x^52

The gray image is a linear code over GF(4) with n=52, k=7 and d=28.
This code was found by Heurico 1.16 in 0.153 seconds.